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Simplify the expression [tex]\( i^{37} \)[/tex].

[tex]\[
\begin{aligned}
i^{37} &= i^{36+1} \\
&= i^{36} \times i \\
&= (i^4)^9 \times i \\
&= 1^9 \times i \\
&= 1 \times i \\
&= i
\end{aligned}
\][/tex]

Use the example as a model. Simplify the expression [tex]\( i^{37} \)[/tex].

[tex]\[
i^{37} = \square
\][/tex]


Sagot :

To simplify the expression [tex]\(i^{37}\)[/tex], we can use the properties of the imaginary unit [tex]\(i\)[/tex] and its powers. Recall the cyclic nature of the powers of [tex]\(i\)[/tex]:

[tex]\[ \begin{aligned} i^1 &= i, \\ i^2 &= -1, \\ i^3 &= -i, \\ i^4 &= 1. \end{aligned} \][/tex]

Notice that [tex]\(i^4 = 1\)[/tex], which means any power of [tex]\(i\)[/tex] that is a multiple of [tex]\(4\)[/tex] will be equal to [tex]\(1\)[/tex]. Because of this, we can reduce the exponent [tex]\(37\)[/tex] modulo [tex]\(4\)[/tex] to simplify the expression.

Let's perform the steps:

1. Reduce the exponent modulo 4:
[tex]\[ 37 \mod 4 = 1. \][/tex]

This tells us that:
[tex]\[ i^{37} \equiv i^1 \pmod{4}. \][/tex]

Thus:
[tex]\[ i^{37} = i^1. \][/tex]

2. Identify the equivalent power of [tex]\(i\)[/tex]:

Based on the calculations above:
[tex]\[ i^{37} = i. \][/tex]

So finally, the simplified expression for [tex]\(i^{37}\)[/tex] is:
[tex]\[ i^{37} = i. \][/tex]

[tex]\(\boxed{i}\)[/tex]

Feel free to ask if you have any further questions regarding simplifying expressions that involve the imaginary unit [tex]\(i\)[/tex]!